What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Try out the lottery that is played in a far-away land. What is the chance of winning?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

An activity making various patterns with 2 x 1 rectangular tiles.

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

These practical challenges are all about making a 'tray' and covering it with paper.

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Investigate the different ways you could split up these rooms so that you have double the number.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?